Olympiad problems

2013 Greece JBMO TST, 3

If $p$ is a prime positive integer and $x,y$ are positive integers, find , in terms of $p$, all pairs $(x,y)$ that are solutions of the equation: $p(x-2)=x(y-1)$. (1) If it is also given that $x+y=21$, find all triplets $(x,y,p)$ that are solutions to equation (1).

PEN H Problems, 37

Tags: modular arithmetic , Diophantine Equations

Prove that for each positive integer $n$ there exist odd positive integers $x_n$ and $y_n$ such that ${x_{n}}^2 +7{y_{n}}^2 =2^n$.

PEN H Problems, 91

Tags: geometry , rectangle , perimeter , Diophantine Equations

If $R$ and $S$ are two rectangles with integer sides such that the perimeter of $R$ equals the area of $S$ and the perimeter of $S$ equals the area of $R$, then we call $R$ and $S$ a friendly pair of rectangles. Find all friendly pairs of rectangles.

PEN H Problems, 1

Tags: Diophantine Equations

One of Euler's conjectures was disproved in the $1980$s by three American Mathematicians when they showed that there is a positive integer $n$ such that \[n^{5}= 133^{5}+110^{5}+84^{5}+27^{5}.\] Find the value of $n$.

PEN H Problems, 36

Tags: Diophantine Equations , pen

Prove that the equation $a^2 +b^2 =c^2 +3$ has infinitely many integer solutions $(a, b, c)$.

2017 Latvia Baltic Way TST, 14

Tags: number theory , Diophantine Equations , diophantine

Can you find three natural numbers $a, b, c$ whose greatest common divisor is $1$ and which satisfy the equality $$ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) ?$$

2014 Junior Balkan MO, 1

Tags: modular arithmetic , Diophantine equation , Diophantine Equations

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

1979 IMO Shortlist, 15

Tags: algebra , system of equations , Diophantine Equations , IMO , Cauchy-Schwarz inequality

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

2018 India PRMO, 6

Tags: algebra , number theory , Sum , Diophantine Equations , diophantine

Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?

PEN H Problems, 30

Tags: Diophantine Equations

Let $a$, $b$, $c$ be given integers, $a>0$, $ac-b^2=p$ a squarefree positive integer. Let $M(n)$ denote the number of pairs of integers $(x, y)$ for which $ax^2 +bxy+cy^2=n$. Prove that $M(n)$ is finite and $M(n)=M(p^{k} \cdot n)$ for every integer $k \ge 0$.

2019 District Olympiad, 1

Tags: algebra , number theory , Diophantine equation , Diophantine Equations

Determine the integers $a, b, c$ for which $$\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}$$

1985 IMO Longlists, 82

Tags: algebra , polynomial , Diophantine Equations

Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

2023 Bangladesh Mathematical Olympiad, P1

Tags: factorial , number theory , Diophantine Equations

Find all possible non-negative integer solution $(x,y)$ of the following equation- $$x! + 2^y =(x+1)!$$ Note: $x!=x \cdot (x-1)!$ and $0!=1$. For example, $5! = 5\times 4\times 3\times 2\times 1 = 120$.

2020 Chile National Olympiad, 4

Tags: number theory , Diophantine equation , Diophantine Equations , diophantine , system of equations

Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$

PEN H Problems, 49

Tags: search , group theory , abstract algebra , Diophantine Equations

Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.

PEN H Problems, 67

Tags: Diophantine Equations

Is there a positive integer $m$ such that the equation \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}= \frac{m}{a+b+c}\] has infinitely many solutions in positive integers $a, b, c \;$?

PEN H Problems, 46

Tags: calculus , integration , Diophantine equation , Diophantine Equations

Let $a, b, c, d, e, f$ be integers such that $b^{2}-4ac>0$ is not a perfect square and $4acf+bde-ae^{2}-cd^{2}-fb^{2}\neq 0$. Let \[f(x, y)=ax^{2}+bxy+cy^{2}+dx+ey+f\] Suppose that $f(x, y)=0$ has an integral solution. Show that $f(x, y)=0$ has infinitely many integral solutions.

2013 Greece JBMO TST, 3

PEN H Problems, 37

PEN H Problems, 91

PEN H Problems, 1

PEN H Problems, 36

2017 Latvia Baltic Way TST, 14

2014 Junior Balkan MO, 1

1979 IMO Shortlist, 15

2018 India PRMO, 6

PEN H Problems, 30

2019 District Olympiad, 1

1985 IMO Longlists, 82

2023 Bangladesh Mathematical Olympiad, P1

2020 Chile National Olympiad, 4

PEN H Problems, 49

PEN H Problems, 67

PEN H Problems, 46

2016 Czech-Polish-Slovak Junior Match, 6

PEN H Problems, 71

PEN H Problems, 86

2021 Serbia Team Selection Test, P3

PEN H Problems, 15

PEN H Problems, 3

PEN H Problems, 61

PEN H Problems, 24